From nuclei to neutron stars: simple binding energy computer modelling in the classroom (Part 1) A Pastore Department of Physics, University of York, Heslington, York, Y010 5DD, United Kingdom E-mail: [email protected] A M Romero Department of Physics and Astronomy, The University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599, USA C Diget Department of Physics, University of York, Heslington, York, Y010 5DD, United Kingdom A Rios Department of Physics, Faculty of Engineering and Physical Sciences, University of Surrey, Guildford, Surrey GU2 7XH, United Kingdom K Leech Department of Physics, University of York, Heslington, York, Y010 5DD, United Kingdom P Stokoe Department of Physics, University of York, Heslington, York, Y010 5DD, United arXiv:2007.06872v2 [nucl-th] 26 Apr 2021 Kingdom Abstract. We present a simple activity based on the liquid-drop model which allows secondary school students to explore the uses of mathematical models and gain an intuitive understanding of the concept of binding energy, and in particular the significance of positive binding energy. Using spreadsheets provided as Supplementary Material, students can perform simple manipulations on the different coefficients of the model to understand the role of each of its five terms. Students can use the spreadsheets to determine model parameters by optimising the agreement with real atomic mass data. This activity can be used as the starting point of a discussion about theoretical models, their validation when it comes to describing experimental data and their predictive power towards unexplored regimes. From nuclei to neutron stars: simple binding energy computer modelling in the classroom (Part 1)2 1. Introduction The Binding Blocks (BB) project is an educational outreach activity developed at the University of York aimed at promoting the knowledge of nuclear science to a large audience and in particular to young studentsz. In Ref. [1], some of us discussed the main outlines of the project. By using different towers of LEGO® bricksx, we create a chart of nuclides in three dimensions. The towers are colour-coded according to the radioactive mechanism through which isotopes decay. The height of each tower represents the mass excess per nucleon per kilogram of each isotope relative to 56Fe, the most stable nucleus. An online version of the nuclear chart has also been created and can be found in Ref. [2]. Using the three dimensional chart, we have also developed a series of activities for A-level students [3]. The primary goal is to use the striking visual impact of the chart as a formative tool for teachers to explain complicated aspects related to nuclear science. On a devoted YouTube® channel, we have also published mini-lectures focusing on nuclear physics using this three-dimensional LEGO® chart k. These efforts have focused on experimental nuclear data [1] or on nuclear processes [4], but have not addressed specifically theoretical nuclear physics ideas. The goal of this paper is to introduce a new series of activities that deal with the concept of a theoretical nuclear model and its applications. We use, as a representative case, a set of relatively simple mathematical equations based on the liquid-drop (LD) model. The LD model, first introduced by George Gamow [5], describes the nucleus as a drop of incompressible fluid of very high density, held together by the nuclear force. Our aim is to demonstrate that models can be used to describe physical data; that models often involve a parameter optimisation process; that they can be exploited to reach conclusions far beyond their initial remit; and that they bring in their own set of (systematic) uncertainties. Theoretical model uncertainties are in fact of a different nature than those addressed in experiments, and can be explored with our proposed activities. In doing so, we expect to trigger a discussion about the fundamentals of scientific models and their applicability [6,7]. In the present activity, we specifically focus on a single nuclear physics concept - that of binding energy (BE). Binding energies are crucial to understand why a nucleus exists, and they can be used to explain why only specific combinations of protons and neutrons are found in nature or in experimental nuclear physics facilities. BEs are routinely measured in nuclear experiments, so there is a wealth of data to compare to. Using a very intuitive theoretical model based on the LD binding energy formula, students can gain insight into the LD model, its merits and applicability. We provide a series of visualisation tools by means of Microsoft Excel® and OpenOffice worksheets. By removing the complexity of the simulation and providing an immediate visual output, z https://www.york.ac.uk/physics/public-and-schools/secondary/binding-blocks/ x LEGO® is a trademark of the LEGO Group of companies which does not sponsor, authorise or endorse the present work. k https://www.youtube.com/channel/UCvIXlFgJyGh4Jle_4_KE2aA From nuclei to neutron stars: simple binding energy computer modelling in the classroom (Part 1)3 students can explore different aspects without mathematical complications. They can, for instance, use the model to explain the trends observed in nuclear data. Importantly, they can apply the model to predict binding energies beyond those that have been measured, thus providing predictions into experimentally unknown regimes. In a follow- up contribution (Part 2), we explore how the model can be used to extrapolate beyond its initial remit by discussing neutron stars [8]. Across the paper, we propose a series of questions and challenges that can be addressed by students. Some of these do not require access to the computational worksheets, and can therefore be addressed directly in a classroom presentation. The worksheet activities require access to a computer, but allow for an interactive, and more active, learning experience. We provide two different worksheets that can be used in different settings. Worksheet_Instructor contains data for several isotopic chains, and can be run in in classrooms and computational laboratories, or as outreach activities in scientific centers with the guidance of teachers, instructors or tutors. Worksheet_Student is designed to be a standalone online activity, that individual students can run in an online setting - be it at home or in a classroom. This can be accompanied by a minimal instruction set and introduced by online means (e.g. a video like the one here{). The article is organised as follows. In Sec. 2, we introduce the concept of binding energy, while in Sec. 3 we present the LD model equations. Section 4 describes simple paramater optimisation strategies for the model. In Sec. 5 we illustrate how to predict the binding energies of not yet measured nuclei. We present our conclusions in Sec. 6. 2. Binding energy The binding energy of a physical system is the amount of energy required to separate any composite system into all of its constituents. The concept of binding energy is ubiquitous in science, and is particularly relevant for understanding the chemical properties of atoms and molecules; the astrophysics of gravitational systems and the physics of nuclei. In the case of a nucleus, the constituents are the individual Z protons and N neutrons, inclusively called nucleons. The mass number of the nucleus is A = N +Z. The binding energy of an isotope of element Z with N neutrons is defined by the difference 2 2 2 BE = Nmnc + Zmpc − MN;Z c ; (1) 2 + 2 where mnc = 939:565 MeV and mpc = 938:272 MeV represent the rest mass energies of the neutron and the proton, and MN;Z is the rest mass of the isotope itself. Reference [9] provides a more detailed discussion and some simple practical activities to familiarise interested readers with the concept of binding. It is important to differentiated binding energies from separation energies, which instead correspond to { https://www.youtube.com/watch?v=Qsu7IrGiOIk + In nuclear physics activities, it is more natural to work in units of mega-electronvolts (MeV) rather than Joules (J). We recall that 1 J=6:242 · 1012 MeV From nuclei to neutron stars: simple binding energy computer modelling in the classroom (Part 1)4 8 56 Fe 6 Oxygen 4 Calcium Iron BE/A (MeV) Lead 2 Fusion Fission 0 0 50 100 150 200 250 300 Mass number, A=N+Z Figure 1: (Color online) Binding energy per nucleon, BE=A in MeV, as a function of mass number A. Data points come from the experimental measurements compiled in Ref. [14]. See text for details. the minimum energy required to remove one (or more, but not all) the constituents of the composite system. The typical binding energy per particle of a medium-to-heavy nucleus is of the order of ≈ 8 MeV. Using advanced experimental techniques [10{12], it is possible to measure the binding energies of very short-lived radioactive nuclei, far away from the valley of stability. According to the most recent nuclear database [13], more than 2400 nuclear binding energies per particle have been measured with an accuracy of more than 0:005%. With such remarkable levels of accuracy, the error bars would not be visible on the figures illustrating experimental values. We have thus decided not to consider the presence of experimental errors in the activities described below. We illustrate the evolution of the experimental binding energy per particle, BE=A, as a function of mass number A in Fig. 1. The black circles correspond to the data for 2411 isotopes from Ref. [14]. On the same plot, we highlight some particular isotopic chains: O (Z = 8), Ca (Z = 20), Fe (Z = 26) and Pb (Z = 82). A number of interesting conclusions can be drawn from this figure alone, which could very well be the starting point for the discussion of the activities we propose below.